On the asymptotic behaviour of the number of subgroups of finite abelian groups
Let \[ t_2(n) = \sum_{| {\mathcal G}| =n,\;r({\mathcal G})\leq 2}\tau({\mathcal G}), \] where \(\tau({\mathcal G})\) denotes the number of subgroups of a finite abelian group \({\mathcal G}, r({\mathcal G})\) is the rank of \({\mathcal G}\), and \(| {\mathcal G}| \) is the order of \({\mathcal G}\). The group \({\mathcal G}\) has rank \(r\) if \[ {\mathcal G} \cong \mathbb Z/n_1\mathbb Z \otimes \cdots \otimes \mathbb Z/n_r\mathbb Z, \] where \(n_j\mid n_{j+1}\) for \(j = 1,\cdots,r - 1\). We set \[ T(x) = \sum_{n\leq x}t_2(n) = \sum_{| {\mathcal G}| \leq x,r({\mathcal G})\leq 2}\tau({\mathcal G}), \] so that one has \[ T(x) = K_1x\log^2x + K_2x\log x + K_3x + \Delta(x), \] where \(K_j\) are effective constants and \(\Delta(x)\) is to be considered as the error term in the asymptotic formula for \(T(x)\). The bound \(\Delta(x) \ll x^{31/43+\varepsilon}\) was proved by \textit{G. Bhowmik} and \textit{H. Menzer} [Abh. Math. Semin. Univ. Hamb. 67, 117-121 (1997; Zbl 0890.11027)]. Further \textit{H. Menzer} [Proc. Conf. analytic number theory (ed. W. G. Nowak), Vienna 1996, Universität Wien, 181-188 (1996; Zbl 0879.11049)] improved the above bound to \(\Delta(x)\ll x^{9/14+\varepsilon}\) by using two estimates in the three-dimensional asymmetric divisor problem. In the present work the authors obtain the bound \(\Delta(x)\ll x^{5/8}\log^4x\). This is achieved by obtaining the same bound, by the intricate technique of multiple exponential sums, for the error term in the asymptotic formula for the summatory function of \(f(n) := \sum_{n=n_1n_2n_3^2}n_3\), where \(n_1,n_2,n_3\) are natural numbers. This result, which is hitherto the sharpest one, is certainly of independent interest. From the shape of the relevant generating function it seems reasonable to expect that, for suitable constants \(A_j\), one has \[ \Delta(x) = A_1x^{1/2}\log^2x + A_2x^{1/2}\log x + A_3x^{1/2} + E(x), \] where \(E(x)\) may be considered as a further error term. The authors prove that, for any given \(\varepsilon > 0\), one has \[ {1\over x}\int_0^x E(u) \text{ d}u \ll_\varepsilon x^{3/8+\varepsilon}, \] from which it follows that \(\Delta(x) = \Omega_-(x^{1/2}\log^2x)\). The paper concludes with two results on the maximal order of \(t_2(n)\) and its comparison to the number of distinct prime divisors of \(n\).
scientific article; zbMATH DE number 1145400